Elimination Method

How to solve a system of linear equations and inequalities by using their graphs: 4 examples and their solutions.

Example 1

Example

Solution

Choose the variable you want to remove.

The same or the opposite terms are easier to remove.
So choose y as the variable to remove.

Make the y terms the same or the opposites.

-y and +y are already the opposites.
So write x - y = 4 and 2x + y = 5.

To remove the y terms,
add the equations.

x + 2x = 3x
The y terms are removed.
4 + 5 = 9

So 3x = 9.

Divide both sides by 3.
Then x = 3.

Put this x = 3
into one of the given equations.

x - y = 4 looks simpler.
So put x = 3 into x - y = 4.

Then 3 - y = 4.

Substitution Method

Find the value of y.
Then y = -1.

So [x = 3, y = -1] is the answer.

Example 2

Example

Solution

Choose the variable you want to remove.

This time, let's choose the variable x to remove.

Make the x terms the same or the opposites.

The x terms are 3x and 2x.
So change 3x and 2x to 6x.

So multiply 2 to the both sides of 3x + 2y = 7:
6x + 4y = 14.

Next, see the other equation 2x - 3y = -4.

To change 2x to 6x,
multiply 3 to the both sides of 2x - 3y = -4:
6x - 9y = -12.

So the given equations are changed to
6x + 4y = 14 and 6x - 9y = -12.

The x terms are the same: 6x.

To remove the x terms,
subtract the equations.

The x terms are removed.
4y - (-9y) = 4y + 9y = 13y
14 - (-12) = 14 + 12 = 26

So 13y = 26.

Divide both sides by 13.
Then y = 2.

Put this y = 2
into one of the given equations.

Put y = 2 into 3x + 2y = 7.

Then 3x + 2⋅2 = 7.

Find the value of x.
Then x = 1.

Linear Equation (One Variable)

So [x = 1, y = 2] is the answer.

Example 3

Example

Solution

Choose the variable you want to remove.

Choose x as the variable to remove.

Make the x terms the same or the opposites.

The x terms are x and 2x.
So, to make the same 2x,
change x to 2x.

So multiply 2 to the both terms of x - y = 4:
2x - 2y = 8.

And write 2x - 2y = 8.

To remove the x terms,
subtract the equations.

The x terms are removed.
The y terms are removed.
8 - 8 = 0

So 0 = 0.

0 = 0 is always true.

Just like this case,
when you get an equation that is always true,
then the system has infinitely many solutions.

So [infinitely many solutions] is the answer.

Example 4

Example

Solution

Choose the variable you want to remove.

Choose x as the variable to remove.

Make the x terms the same or the opposites.

The x terms are both the same x.

So write x - y = 4 and x - y = -3.

To remove the x terms,
subtract the equations.

The x terms are removed.
The y terms are removed.
4 - (-3) = 4 + 3 = 7

So 0 = 7.

0 = -7 is always false.

Just like this case,
when you get an equation that is always false,
then the system has no solution.

So [no solution] is the answer.